A Feasibility-Preserving Local Search Operator for ... - CiteSeerX

This is the author’s version of the work. The definitive work was published in Proceedings of the 2008 IEEE Congress on Evolutionary Computation (CEC 2008), pp. 1968-1975, 2008. The work is supported in part by the German Science Foundation (DFG), SFB 694.

A Feasibility-Preserving Local Search Operator for Constrained Discrete Optimization Problems Martin Lukasiewycz, Michael Glaß, Christian Haubelt, and J¨urgen Teich

Abstract— Meta-heuristic optimization approaches are commonly applied to many discrete optimization problems. Many of these optimization approaches are based on a local search operator like, e.g., the mutate or neighbor operator that are used in Evolution Strategies or Simulated Annealing, respectively. However, the straightforward implementations of these operators tend to deliver infeasible solutions in constrained optimization problems leading to a poor convergence. In this paper, a novel scheme for a local search operator for discrete constrained optimization problems is presented. By using a sophisticated methodology incorporating a backtracking-based ILP solver, the local search operator preserves the feasibility also on hard constrained problems. In detail, an implementation of the local serach operator as a feasibility-preserving mutate and neighbor operator is presented. To validate the usability of this approach, scalable discrete constrained testcases are introduced that allow to calculate the expected number of feasible solutions. Thus, the hardness of the testcases can be quantified. Hence, a sound comparison of different optimization methodologies is presented.

I. I NTRODUCTION

M

ANY meta-heuristic optimization approaches rely on a unary local search operator. In the domain of Simulated Annealing [1] and Tabu Search [2] this operator is known as the neighbor operator. A straightforward implementation of a neighbor operator on discrete problems is, e.g., a single bit-flip. Other optimization approaches like, e.g., Evolution Strategies [3] are based on a mutate operator. In this case, a common mutate implementation flips a single bit with a given probability. Hence, the common ground of these operators is the variation of a given solution to a specific amount. For discrete problems with linear constraints it can be NPcomplete to find a single feasible solution [4]. In general, for these constrained optimization problems, a local search for a varied feasible solution is NP-complete, too. Thus, it might appear that a straightforward implementation of the local search operator very often results in infeasible solutions. Hence, many approaches in optimization heuristics rely on local repair strategies whereas infeasible solutions are accepted. By using penalty functions, the objectives are deteriorated to guide the search toward the feasible solutions. However, in hard constrained problems, these approaches tend to fail since they are more focused on the search for feasible solutions than on the optimization of the objectives. In this paper, a novel scheme for a local search operator for linear discrete constrained optimization problems that Hardware/Software Co-Design, Department of Computer Science, University of Erlangen-Nuremberg, Germany, {martin.lukasiewycz, glass, haubelt, teich}@cs.fau.de

obtains feasible solutions only is presented. This scheme for local search operators incorporates a Pseudo-Boolean (PB) solver [5], [6], a specialized backtracking-based Integer Linear Program (ILP) with binary variables. In detail, an implementation of a neighbor operator in the context of a Simulated Annealing and a mutate operator in an Evolution Strategy is presented. To validate this methodology, scalable testcases are introduced. These testcases are scalable in the number of constraints and variables following a defined constructions procedure. This procedure allows a calculation of the average ratio of feasible solutions to the overall number of solutions as well as the expected absolute number of feasible solutions per testcase. Thus, the hardness of a testcase can be quantified and a sound comparison of the proposed local search operators to known methodologies is achieved, respectively. The remainder of the paper is outlined as follows: Section II gives a short introduction of related work and Section III defines the problem of constrained discrete optimization. In Section IV, the feasibility-preserving local search operator is presented. Section V introduces scalable testcases for constrained discrete optimization. Experimental results are given and discussed in Section VI before the paper is concluded in Section VII. II. R ELATED W ORK For discrete problems, it is common to use heuristic optimization approaches, mainly if the problem is too large to be solved completely or if it has multiple or non-linear objectives. However, the heuristic optimization of constrained discrete problems is still mainly unexplored. General methods for constraint handling are outlined in [7], [8], [9] and are summarized in the following: A common method is the usage of penalty functions [7]. Depending on the number of unsatisfied constraints, a penalty value is added to the objective functions and, thus, the fitness of the individual is deteriorated. Thereby, feasible solutions and solutions with low penalty values are prioritized automatically in the optimization process. Prioritizing feasible solutions over infeasible by a clear distinction is a well known strategy [10], too. For some problems, the feasibility of solutions can be preserved by eliminating decision variables or using specialized operators [11]. Furthermore, there exist local repair strategies which based on additional domain-specific information to fix an infeasible solution [7], [11]. However, repair strategies that are restricted to a local view are not able to guarantee to deliver feasible solutions only.

Known methods for constrained handling are either restricted to real valued search spaces or tend to fail if the discrete problem is hard constrained since the optimization is more focused on the search for feasible solutions than optimizing the objectives. In [12], a decoding strategy is presented that overcomes these drawbacks, by using a PB solver as a decoder. Thus, only feasible solutions are obtained within the search process of an Evolutionary Algorithm. This decoder-based approach can focus on the optimization of the objectives and a good convergence towards the optimal solutions is reached. In this paper, an alternative methodology that uses a PB solver as a local search operator is presented instead. By using a PB solver on discrete constrained problems as a mutate or neighbor operator, only feasible solutions are obtained. Thus, the proposed search processes overcome the drawbacks of traditional straightforward local search operators which often stick to infeasible regions in the search space prohibiting the convergence towards optimal solutions. To evaluate the performance of the proposed methodology, appropriate testcases are needed. Common benchmarks for discrete constrained optimization with meta-heuristic approaches are, e.g., the 0/1 Knapsack Problem [13] or the Traveling Salesman Problem [14]. In fact, obtaining a feasible solution for these problems is trivial. With domainspecific knowledge, it is a common method to apply a straightforward repair algorithm [13] or operators that preserve the feasibility of the solution [14], respectively. In [12] and [15], also problems where obtaining one feasible solution is NP-complete are created. However, these testcases are not scalable in the degree of hardness. On the other hand, there exists a set of handmade and industrial benchmarks for the optimization of Integer Linear Programs with binary variables [16]. Though, these could be easily extended to multiple and also non-linear objectives, none of these testcases gives information about the difficulty to obtain one feasible solution since these benchmarks are not used for meta-heuristic approaches but for complete ILP and PB solvers. In this paper, a construction procedure for scalable testcases is introduced. By using this construction procedure, the calculation of the average ratio of feasible solutions to the size of the complete search space is enabled. These testcases are scalable in the number of variables and constraints. Thus, a sound comparison of the proposed methodology is obtained.

(a)

(b)

Fig. 1. Illustration of different local search operators in the search space of a constrained problem. The circles are feasible solutions and the arrows show one possible local search operation. (a) illustrates a straightforward local search operator that results in infeasible solutions only. (b) is a feasibilitypreserving local search operator.

Xf ⊆ X is defined as the set of all solutions x that fulfill all constraints. Though the demand that all constraints have to be linear with binary variables sounds rather restrictive, a linearization of products of binary variables is allowed by introducing additional variables and constraints 1 . Moreover, integer variables can be encoded binary. Given the definition of the feasible search space Xf and the objective function f , the Constrained Combinatorial Optimization is defined as the following multi-objective optimization problem: Definition 1 (Constrained Combinatorial Optimization) optimize f (x) subject to: x ∈ Xf

(1)

The objective function f consists of multiple functions including also non-linear calculations. In a single-objective optimization, the set of feasible solutions is totally ordered, whereas in multi-objective optimization problems, the feasible set is only partially ordered and, thus, there is generally not only one global optimum, but a set of Pareto solutions. A Pareto-optimal solution is not worse or equal in all objectives than any other feasible solution in the design space [17]. In hard constrained problems the set of feasible solutions Xf is significantly smaller than X and the search space contains a high number of infeasible solutions, respectively. If there exists no domain specific repair algorithm or feasibility-preserving operator, a straightforward local search tends to obtain infeasible solutions mainly. This problem is illustrated in Fig. 1(a). For optimization problems like the Minimal Hamiltonian Circuit and the optimization version of the Exact Cover Problem [4] as well as the industrial application of Design Space Exploration [18] it is a NPcomplete problem to obtain a single feasible solution and can be formulated as a set of linear constraints with binary variables. In this paper, a procedure for a construction of testcases where obtaining a single feasible solution is NPcomplete is presented. Moreover, this construction procedure allowes to formalize the hardness of this problem as the

with a ∈ Zn , b ∈ Z and ◦ ∈ {}. Thus, by introducing a set of constraints, the feasible search space

1 linearization substitution rule: x · x ↔ x with x − x ≥ 0 ∧ x − 1 2 3 1 3 2 x3 ∧ x1 + x2 − x3 < 2

III. P ROBLEM F ORMULATION For constrained discrete problems the task of metaheuristic optimization approach is twofold: (1) The objectives have to be optimized whereas (2) all constraints have to be fulfilled. In this paper, all variables are assumed to be binary variables. Thus, the search space is X = {0, 1}n . All constraints are linear, with a single constraint formulated as aT x ◦ b

Algorithm 1 DPLL backtracking algorithm 1: while true do 2: branch(ρ, σ) 3: if CONFLICT then 4: backtrack() 5: else if SATISFIED then 6: return x 7: end if 8: end while

expected number of feasible solutions or the average ratio of feasible solution. With a decreasing ratio of feasible to infeasible solutions, a straighforward local search operator obtains a growing number of infeasible solutions. A meta-heuristic approach that uses a naive operator on this class of prolems tends to obtain mainly infeasible solutions. Therefore, these approaches are more focused on the search for feasible solutions than on the optimization of the objectives. Thus, a local search operator that only delivers feasible solutions is desired. Such an operator is illustrated in Fig. 1(b). The illustrated approach preserves the feasibility of solutions regardless of the degree of hardness of the problem. Moreover, this methodology is not domain-specific and can be applied to any problem where obtaining one feasible solution can be encoded into binary linear constraints. IV. F EASIBILITY- PRESERVING L OCAL S EARCH O PERATOR This section presents a scheme for a feasibility-preserving local search operator for linear constrained discrete problems. This scheme incorporates a Pseudo-Boolean (PB) solver [5], [6]. A PB solver uses a backtracking algorithm to solve Integer Linear Programs with an empty objective function and binary variables. The introduced scheme is used to implement a feasibility-preserving neighbor and mutate operator. Moreover, these operators can be used to effectively reduce the search space within the search process since these operators recognize if some variables have to be assigned a specific value to obtain a feasible solution. A. PB Solver The task of a PB solver is to find an x ∈ {0, 1}n that satisfies a set of linear constraints. In fact, this NP-complete problem [4] is an ILP with binary variables and an empty objective function that can be solved by a common ILP solver. However, due to the boolean nature, the specialized PB solvers tend to outrun common ILP solvers on these problems [19]. These PB solvers are extended SAT solvers that are actually used to solve the Satisfiability problem and are based on a backtracking strategy. This strategy is known as the DPLL algorithm [20] and is outlined in Algorithm 1. The algorithm efficiently searches for a solution x that fulfills all constraints, cf. [5], [6]: Starting with completely unassigned variables, the operation branch(ρ, σ) chooses an unassigned variable and assigns it a value (line 2). The branching rule which variable is chosen and value is assigned

Operator O x

deduce branching strategy

σ, ρ

solve

x'

Fig. 2. PB solver based local search operator. Given is the solution x ∈ Xf . Following x, a branching strategy ρ, σ for a PB solver is deduced. With this branching strategy, the PB solver finds an feasible x0 ∈ Xf automatically.

is called decision strategy. Decision strategies are often guided by two vectors ρ ∈ Rn and σ ∈ {0, 1}n . Unassigned variables xi with the highest value ρi are prioritized and are set to the value σi . The configuration of the vectors ρ and σ is called branching strategy since it determines the backtracking branching process. After each variable assignment, possible conflicts are recognized (line 3). That means, if any constraint is not satisfiable anymore, the backtracking is triggered (line 4), i.e., variable assignments are reverted. In the case that all variables have an assignment and there exists no conflict (line 5), a feasible solution x is found and returned (line 6). B. Feasibility-preserving Operator Scheme The general task of a feasibility-preserving local search operator is: For a given solution x ∈ Xf to find a feasible solution x0 ∈ Xf . This local search operator O : Xf → Xf is illustrated in Fig. 2. The function O is twofold: First, a branching strategy σ, ρ is deduced from the solution x. Second, the PB solver generates a feasible solution x0 using the deduced branching strategy. In fact, the deduced branching strategy has a huge impact on the found solution. For a given σ and two feasible solutions x0 , x00 ∈ Xf it holds: The solution x0 is more similar to σ than x00 if |{i|x0i = σi }| > |{i|x00i = σi }|.

(2)

Now, given a σ and random priorities ρ, the probability that x0 is found by a backtracking strategy in a PB solver is higher than the probability that x00 is found. This is due to the fact that with Equation (2) it holds |{i|x0i 6= σi }| < |{i|x00i 6= σi }|.

(3) 00

Thus, with randomly assigned priorities, x is excluded with a higher probability early in the backtracking search process compared to x0 . Hence, x0 is reached with a higher probability than x00 . Moreover, in a special case where for all x00i = σi holds that x0i = σi , the solution x00 is not reachable at all due to the presence of x0 . Thus, the backtracking strategy tends to find an x0 that is more similar to σ than other feasible solutions. This holds under the assumption that the priorities are random values. Thus, the vector σ allows to influence the found solution of a PB solver based local search operator, i.e., the similarity

x x1 . . xi . . xn

Neighbor N

i

σ1=x1 . .

ρ1=r1 . .

σi=xi . .

ρi=h+ri . .

σn=xn

ρn=rn

σρ

solve

x'

x

Mutate M

x'

x1' . .

x1 ..

σ1=x1 ..

x1' .. xi' . xj' .. xn'

xi' . . xn'

xi . xj .. xn

i,j,... є I

ρ1=r1 ..

σi=xi ρi=h+ri . . σj=xj ρj=h+rj .. .. σn=xn ρn=rn

σρ

solve

Fig. 3. Feasibility-preserving Neighbor Operator. One bit i is forced to flip with a high priority.

Fig. 4. Feasibility-preserving Mutate Operator. A certain amount of bits I are prioritized to flip. This set I deduced by the mutation rate.

of x and x0 can be influenced by the way σ is deduced from x. This is utilized by the operators that are introduced in the following.

D. Feasibility-preserving Mutate Operator Mutate operators are applied to many meta-heuristic optimization methods. The Evolution Strategy is a popular methodology that relies on a mutate operator merely. In fact, a mutate operator has several similarities to the neighbor operator: Given a solution x ∈ X, the task of the mutate operator is to find an x0 ∈ X. In a constrained optimization both solutions are aspired to be feasible. Thus, the feasibilitypreserving mutate operator M : Xf → Xf is implemented following the proposed operator scheme. A good mutate operator fulfills the following condition: 1) The difference between x and x0 are of a distinct degree. The novel feasibility-preserving mutate operator M for discrete constrained optimization problems is illustrated in Fig. 4. In a first step, a set I ⊆ {1, ..., n} is selected by adding each element with a probability of the mutation rate. With a growing mutation rate the size of the set I tends to increase. The mutate operator sets for each variable i ∈ I the corresponding σi to xi , thus, prioritizing a mutation of these bits. Therefore, the priority ρi for the bit i ∈ I is increased by the large number h (maxi ri + 1) to ensure the mutation of these variables. Correspondingly, for each bit i ∈ / I the σi is set to xi . Like in the neighbor operator, the priorities are randomly selected from R. With a growing size of I, the number of forced bit flips increases and, thus, the resulting solutions x0 tends to have a greater difference to x. In fact, the neighbor operator is a special case of the mutation operator with |I| = 1. In case the problem is unconstrained, this operator converges to the standard mutate operator that flips a single bit with a given probability.

C. Feasibility-preserving Neighbor Operator The most common optimization heuristics that are using a neighbor operator are Simulated Annealing and Tabu Search. Given a solution x ∈ X, the task of the neighbor operator is to find a neighbor x0 ∈ X. In a constrained optimization, it is aspired that both solutions are feasible. With the scheme defined in the previous section, the proposed neighbor operator N : Xf → Xf is realized by a PB solver that preserves the feasibility of the solutions. A good neighbor operator fulfills two conditions: 1) The neighbor is never equal to the original solution, i.e., x0 6= x. 2) The difference between the solutions x and x0 is aspired to be as small as possible. The novel feasibility-preserving neighbor operator N for constrained discrete optimization problems is illustrated in Fig. 3. In the deduced branching strategy, at least one bit flip is forced by randomly selecting one i ∈ {1, ..., n} and setting the corresponding σi to xi (the negated value of xi ). The priority of the i-th bit is increased by a large number h (maxi ri + 1) to ensure that the PB solver flips the ith bit. The remaining σ values are inherited from x such that the branching strategy tends to find a solution x0 that is to a large extend similar to x. Thus, the priorities within the branching strategy have to be randomly selected from a uniform distribution, i.e., each ρ value is a random number r ∈ R. This random priority assigment is done to fulfill the assumption from the previous section such that a near neighbor is found. In the backtracking search process, first the i-th variable is set to xi . In the following, the PB solver tries to find an assignments for the remaining variables that fulfills all constraints. Since the remaining σ is inherited from x, the search tends to find a similar solution. In case the problem is unconstrained, no conflict will occur. Thus, this operator converges to the standard neighbor operator that flips a single bit since x0 equals σ.

E. Search Space Reduction The presented operators always obtain feasible solutions. Under some circumstances, this allows the search space reduction of an optimization heuristic that relies on these operators at runtime. For instance, if the neighbor operator selects the i-th bits to be set to xi with a very high priority, the PB solver is forced to find such a solution. But, if there exists no solution where is the i-th bit is xi , the PB solver will return a x0 that equals x. On one hand, this violates the condition that the neighbor x0 is not allowed to be equal to

x. On the other hand, in the further optimization process the i-th bit can be ignored since there only exists solutions with xi . Thus, the search space can be reduced efficiently within the optimization process. A general rule for the reduction of the search space is as following: Definition 2 (Search Space Reduction Rule) For a given set of I bits that are prioritized to be flipped and a PB solver based feasibility-preserving operator O : Xf → Xf , it holds: If the solution x0 = O(x) equals the original solution x, the bits in I can be removed from the search space. The variables that are removed from the search space are set to the values from the solution x or x0 , respectively, for the following solutions. For the neighbor operator this rule applies to I = {i}. V. S CALABLE T ESTCASES This section presents scalable testcases for discrete constrained optimization problems where obtaining a single feasible solution is an NP-complete problem. First, the construction procedure for these testcases that are scalable in the number of variables and constraints are introduced. Moreover, a determination of the hardness of the testcases is presented. A. Constraint Construction Scheme The construction of the constraints for the scalable testcases relies on three variables: • n ∈ N - number of binary variables • c ∈ N - number of constraints c • m ∈ N with m ∈ N - occurrence of each variable among the constraints The constraints of this problem are generated as follows: Exactly c empty sets S1 , ..., Sc of variables are created. Each variable is randomly distributed over these sets such that each variable occurs in exactly m sets. For each set Si ∈ {S1 , ..., Sc } a constraints of the form X x=1 x∈Si

is created. It has to be ensured that each set S is nonempty to allow a feasible solution. Under the assumption c m ∈ N and each variable occurs exactly m times among the constraints, it is obvious that if a solution exists, it has c c exactly m variables set to 1 and n − m variables set to 0. Moreover, this problem equals Exact Cover Problem that is known to be NP-complete. The strict construction procedure allows to calculate the expected number of feasible solutions |Xf |, i.e., the cardinality of the set Xf , for one testcases (for Proof see Appendix): Theorem 1 c   (c − m)! m n E(|Xf |) = c · c m c! m −1

testcase TC1a TC1b TC2a TC2b

n 200 200 300 300

c 80 30 120 60

m 2 3 2 3

E(|Xf |) 1.37 · 109 8.09 · 104 3.15 · 1013 3.52 · 106

p(x ∈ Xf ) 8.54 · 10−52 5.03 · 10−56 1.54 · 10−77 1.72 · 10−84

TABLE I PARAMETERS FOR THE TESTCASES : T HE NUMBER OF VARIABLES c, THE NUMBER OF CONSTRAINTS n, AND THE NUMBER OF OCCURENCES OF EACH VARIABLE WITHIN THE CONSTRAINTS m. A LSO GIVEN IS THE NUMBER OF EXPECTED FEASIBLE SOLUTIONS E(|Xf |) AND THE AVERAGE RATIO OF FEASIBLE SOLUTIONS TO THE SIZE OF THE SEARCH SPACE

p(x ∈ Xf ).

Thus, the average ratio of feasible solutions is calculated as following: p(x ∈ Xf ) =

E(|Xf |) E(|Xf |) = |X| 2n

Using this ratio of feasible solutions to the size of the search space allows the quantification of the hardness of the testcase. B. Objective Function The objective functions can be selected indepently from the constraints. Here, the objective functions are linear since the focus of this paper is on constraint handling techniques. Each single objective function is f (x) =

n X

ri · xi

i=1

with ri ∈ N the randomly generated in [0; 100]. Non-linear objective functions like NK landscapes [21] are applicable, too. VI. E XPERIMENTAL R ESULTS In this section, experimental results are presented. After an introduction of the used testcases, the proposed neighbor and the proposed mutate operator are analyzed. Afterwards, the performance of these operators using a Simulated Annealing and an Evolution Strategy approach are compared to a stateof-the-art SAT-decoding based Evolutionary Algorithm presented in [12]. The section is concluded with an application of the proposed methodology to a real-world example from the automotive network area. The experimental results are based on an implementation of the local search operators using the PB solver SAT4J [22]. All test cases were carried out on an Intel Pentium 4 3.20 GHz machine with 1GB RAM. A. Testcases Four testcases with 10 instances per testcase were created following Table I. The size of the search space for TC1a and TC1b is 2200 and for TC2a and TC2b 2300 . For each testcase, a single-objective and a multi-objective (two objective functions) problem is generated.

Runtimes[s] of the single-objective optimization:

0.3

neighbor random

avg. difference

0.25

EA (SAT-decoding) ES (Mutate Operator) SA (Neighbor Operator)

0.2 0.15

EA (SAT-decoding) ES (Mutate Operator)

0.05 0 TC1a

TC1b TC2a testcase

TC2b

Fig. 5. Comparison of the difference for the neighbor operator as well as two random solutions.

avg. difference

0.2

TC1b 89.3 (6.3) 108 (9.9) 96.0 (13.4)

TC2a 70.2 (4.1) 65.6 (8.5) 36.6 (1.9)

TC2b 673 (101) 2466 (564) 2723 (511)

Runtimes[s] of the multi-objective optimization:

0.1

0.25

TC1a 58.4 (2.9) 61.1 (7.0) 24.2 (0.4)

TC1a TC1b TC2a TC2b

0.15 0.1 0.05 0 0.001

0.005

0.01 0.05 mutation rate

0.1

0.5

Fig. 6. Influence of the mutation rate on the difference that is induced by the mutate operator.

B. Operator Analysis For the analysis of the proposed novel operators, the normalized difference metric is used. Given two solutions x, x0 ∈ X the difference is: Pn i=1∧xi =x0i 1 0 D(x, x ) = n A small value indicates that x and x0 are similar, a big value that these solution are correspondingly different. All results in the following are an average of 1000 operator calls. In Fig. 5, the difference of two solutions induced by the neighbor operator is stated. To enable a sound comparison, the difference of pairwise random solutions is given, too. In TC1a and TC2a, the difference induced by the neighbor operator is clearly smaller than the random value and, thus, the neighbor operator works as expected. On the other hand, TC1b and TC2b have a sparse neighborhood leading to a comparatively small difference distinction to the pairwise random solutions. Figure 6 shows the influence of the mutation rate on the difference that is induced by the proposed mutate operator. As expected, all testcases show a growing difference with an increasing mutation rate. Correspondingly to the results of the neighbor operator, this influence is more apparent on the testcases TC1a and TC2a. C. Optimization Analysis The novel constraint handling techniques are compared to the Evolutionary Algorithm based SAT-decoding, the stateof-the-art methodology for constrained discrete problems

TC1a 57.2 (2.2) 50.9 (0.3)

TC1b 86.5(3.5) 107 (4.3)

TC2a 72.9 (1.4) 58.3 (4.1)

TC2b 830 (124) 2424 (569)

TABLE II O PTIMIZATION RUNTIMES OF THE COMPARED METHODS . G IVEN IS THE AVERAGE RUNTIME SECONDS AND THE STANDARD DEVIATION IN THE BRACKETS .

with linear constraints [12]. For the SAT-decoding, the used Evolutionary Algorithm (EA) was the elitist SPEA2 [23] algorithm. The population size was set to 100, and in each generation, 25 offspring were created from 25 parents by using crossover and mutation operators with the operators proposed in [12]. To validate the neighbor operator, we used a standard Simulated Annealing (SA) approach [1]. The cooling schedule is linear, starting from a temperature of 100 and ending at 0. The SA is applied to the single-objective optimization only. The presented mutate operator is tested with an (λ + µ) Multi-Objective Evolution Strategy (ES) with λ = 25 and µ = 25 [3]. The mutation mechanism is done by the 1/5 rule. On all handmade testcases the number of evaluations for all methods is set to 25000. In order to evaluate the quality of the methods, the dominance [17] criterion is used. This measurement is used to specify the convergence of multi-objective optimization methods to the front of Pareto-optimal solutions. The dominance calculates the relation of a set of solutions A to the set of the Pareto-optimal solutions B, which is approximated by the best solutions found by all methods in all runs. D (A, B) = inf {b ∈ B | ∃a ∈ A : a  b} 

Thus, the -dominance is the smallest value  with that a set of Pareto-optimal solutions has to be scaled in order to be weakly dominated by the set A. The scaling is normalized and, in the following, the reciprocal value is used such that a high value is aspired. Table II shows the runtimes of the compared methods. In TC1a and TC2a the runtimes of the SA approach with the proposed neighbor operator is significantly lower. This is due the comparatively close neighborhood on these testcases. In contrast, in particularly on TC2b the ES and SA are significantly slower than the EA SAT-decoding approach due to the large search space and sparse neighborhood. Figure 7 shows the results of the single-objective optimization. The SA approach shows the best convergence on all testcases except TC2b. In TC2b, the EA SAT-decoding is significantly better than the proposed local search operator based approaches. In fact, this can be explained by the

1.02

0.25

Evolutionary Algorithm (SAT-Decoding) Evolution Strategy (Neighbor Operator) Simulated Annealing (Mutate Operator)

real-world example

0.2 avg. difference

1 / epsilon-dominance

1.04

1 0.98 0.96

0.15 0.1 0.05

0.94 0 0.001

0.92

0.005

0.01 0.05 mutation rate

0.1

0.5

0.9 TC1a

Fig. 7.

TC1b TC2a testcase

TC2b

Fig. 9. Influence of the mutation rate on the difference that is induced by the mutate operator.

Results of the single-objective optimization. 0.95 Evolutionary Algorithm (SAT-Decoding) Evolution Strategy (Mutate Operator)

1.05 1 / epsilon-dominance

EA (SAT-Decoding) ES (Mutate Operator)

0.9 1 / epsilon-dominance

1.1

1 0.95 0.9

0.85

0.8

0.75

0.7

0.85

0 0.8 TC1a

TC1b

TC2a

TC2b

testcase

Fig. 8.

Fig. 10.

2500 evaluations

5000

Results of the real-world example optimization.

Results of the multi-objective optimization.

sparse neighborhood and, thus, these operators are tending to perform similar to a random search. On the other hand, though TC1b has a sparse neighborhood, too, this search is more successful since the average number of feasible solutions is only about 3-4 times higher than the number of evaluations. Figure 8 shows the results of the multi-objective optimization. The ES optimization based on the proposed mutate operator is better on all testcases compared to the EA based approach. The clearly improved results on TC2b can be explained by a weak performance of the SAT-decoding that leads to a premature convergence. Compared to that, the ES steadily exploits the search space. D. Real-World Example Finally, a real-world example is used to show the practicability of the proposed local search operator. The real-world example is from the automotive network area and consists of 19877 constraints with 7731 binary variables. This example has two objectives, the first is linear (costs) and the second is non-linear (average response time). Thus, the proposed mutate operator within an Evolution Strategy is used and compared to the EA-based SAT-decoding. First, the mutate operator is analyzed on this problem. Figure 9 shows the results. The mutate operator works as expected on this problem as the average difference increases with a growing mutation rate.

Each method was started 3 times such that a sound average value is calculated. Each optimization run was aborted after 5000 evaluations. The average runtime of the ES was 1260 seconds and 1440 seconds for the EA. Figure 10 shows the corresponding results. Thus, the novel mutate operator in combination with the ES shows a better convergence compared to the EA-based SAT-decoding. Moreover, the novel mutate operator within the ES is faster than the decoding strategy with the EA. VII. C ONCLUSION In this paper, a novel scheme for a local search operator for constrained discrete optimization problems as well as scalable testcases that allow to calculate the expected number of feasible solutions are introduced. Based on this scheme, a feasibility-preserving neighbor and mutate operator are presented and applied to a Simulated Annealing and an Evolution Strategy, respectively. Hereby, a branching strategy for a PB solver is deduced from a given feasible solution and, within a backtracking process, a varied feasible solution is found. With a decreasing hardness of the problem, the presented operators converge to a straightforward implementation of the neighbor and mutate operator, respectively. Thus, these operators are applicable regardless of the hardness of the problem that is defined by the constraints. The proposed scalable testcases allow a sound comparison of the presented methodology to former approaches since their hardness can be estimated by the average ratio of

feasible to infeasible solutions. The presented handmade testcases validate the competitiveness of the novel local search operators compared to the state-of-the-art SAT-decoding based Evolutionary Algorithm. Moreover, an application of an Evolution Strategy using the proposed mutate operator performed better than the SAT-decoding approach on a realworld example from the automotive network area. A PPENDIX Proof: In the following, two variables are named disjunctive if they do not appear in the same constraints. The probability that one variable is disjunctive to k other variables (that are pairwise disjunctive) calculates to:

p(k) =

m−1 Y i=0

c−k·m−i with 0 ≤ k < c−i

c m

It is obvious, that each feasible solution x ∈ Xf contains c c exactly m variables that are 1 and n − m variables that are 0. A set that fulfills this requirement is named Xp in c the following with Xf ⊆ Xp ⊆ X. If m randomly chosen variables are disjunctive, these variables can be set to 1 and a feasible solution is found. Thus, the probability for a feasible solution of a random x ∈ Xp is:

p(x ∈ Xf |x ∈ Xp ) =

c −1 m Y

p(k) =

c −1 m Y m−1 Y k=0 i=0

k=0

c−k·m−i c−i

The denominator can be simplified as following: c −1 m Y m−1 Y k=0 i=0

c

−1

m Y 1 = c−i

1

=

c! k=0 (c−m)!

1 c c! m (c−m)!

The numerator can be simplified as following: c −1 m Y m−1 Y

c−k·m−i=

k=0 i=0

c−1 Y

c − j = c!

j=0

An by combination: c

p(x ∈ Xf |x ∈ Xp ) =

c! c c! m (c−m)!

=

(c − m)! m c

c! m −1

With Bayes’ theorem, the number of expected feasible solutions is: c   n (c − m)! m E(|Xf |) = |Xp | · p(x ∈ Xf |x ∈ Xp ) = c · c m c! m −1

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